LMFDB - Embedded newform 6240.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.8266508613$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +2.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.828427 q^{29} -1.65685 q^{31} +4.00000 q^{33} +2.82843 q^{35} -3.65685 q^{37} +1.00000 q^{39} +11.6569 q^{41} +1.65685 q^{43} +1.00000 q^{45} -9.65685 q^{47} +1.00000 q^{49} +0.828427 q^{51} -9.31371 q^{53} -4.00000 q^{55} +2.82843 q^{57} -4.00000 q^{59} -2.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{69} -8.00000 q^{71} +3.17157 q^{73} -1.00000 q^{75} -11.3137 q^{77} -5.65685 q^{79} +1.00000 q^{81} -15.3137 q^{83} -0.828427 q^{85} -0.828427 q^{87} +3.65685 q^{89} -2.82843 q^{91} +1.65685 q^{93} -2.82843 q^{95} -1.51472 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} - 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 8 q^{55} - 8 q^{59} - 4 q^{61} - 2 q^{65} - 16 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 8 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 8 * q^31 + 8 * q^33 + 4 * q^37 + 2 * q^39 + 12 * q^41 - 8 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 - 8 * q^55 - 8 * q^59 - 4 * q^61 - 2 * q^65 - 16 * q^71 + 12 * q^73 - 2 * q^75 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	−1.65685	−0.297580	−0.148790	−	0.988869i	$-0.547538\pi$
$31$	−1.65685	−0.297580	−0.148790	+	0.988869i	$0.547538\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	11.6569	1.82049	0.910247	−	0.414065i	$-0.135891\pi$
$41$	11.6569	1.82049	0.910247	+	0.414065i	$0.135891\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.828427	0.116003
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	3.17157	0.371205	0.185602	−	0.982625i	$-0.440576\pi$
$73$	3.17157	0.371205	0.185602	+	0.982625i	$0.440576\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−11.3137	−1.28932
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.3137	−1.68090	−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137	−1.68090	−0.840449	+	0.541891i	$0.817709\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	−0.828427	−0.0888167
$88$	0	0
$89$	3.65685	0.387626	0.193813	−	0.981039i	$-0.437915\pi$
$89$	3.65685	0.387626	0.193813	+	0.981039i	$0.437915\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	1.65685	0.171808
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−1.51472	−0.153796	−0.0768982	−	0.997039i	$-0.524502\pi$
$97$	−1.51472	−0.153796	−0.0768982	+	0.997039i	$0.524502\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	6.48528	0.645310	0.322655	−	0.946517i	$-0.395425\pi$
$101$	6.48528	0.645310	0.322655	+	0.946517i	$0.395425\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0	0
$111$	3.65685	0.347093
$112$	0	0
$113$	−14.4853	−1.36266	−0.681330	−	0.731976i	$-0.738598\pi$
$113$	−14.4853	−1.36266	−0.681330	+	0.731976i	$0.738598\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−2.34315	−0.214796
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−11.6569	−1.05106
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.34315	0.207921	0.103960	−	0.994581i	$-0.466849\pi$
$127$	2.34315	0.207921	0.103960	+	0.994581i	$0.466849\pi$
$128$	0	0
$129$	−1.65685	−0.145878
$130$	0	0
$131$	−10.8284	−0.946084	−0.473042	−	0.881040i	$-0.656844\pi$
$131$	−10.8284	−0.946084	−0.473042	+	0.881040i	$0.656844\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	13.3137	1.13747	0.568733	−	0.822522i	$-0.307434\pi$
$137$	13.3137	1.13747	0.568733	+	0.822522i	$0.307434\pi$
$138$	0	0
$139$	−2.34315	−0.198743	−0.0993715	−	0.995050i	$-0.531683\pi$
$139$	−2.34315	−0.198743	−0.0993715	+	0.995050i	$0.531683\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	0.828427	0.0687971
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−15.6569	−1.28266	−0.641330	−	0.767265i	$-0.721617\pi$
$149$	−15.6569	−1.28266	−0.641330	+	0.767265i	$0.721617\pi$
$150$	0	0
$151$	20.9706	1.70656	0.853280	−	0.521453i	$-0.174610\pi$
$151$	20.9706	1.70656	0.853280	+	0.521453i	$0.174610\pi$
$152$	0	0
$153$	−0.828427	−0.0669744
$154$	0	0
$155$	−1.65685	−0.133082
$156$	0	0
$157$	21.3137	1.70102	0.850510	−	0.525960i	$-0.176294\pi$
$157$	21.3137	1.70102	0.850510	+	0.525960i	$0.176294\pi$
$158$	0	0
$159$	9.31371	0.738625
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	16.9706	1.32924	0.664619	−	0.747183i	$-0.268594\pi$
$163$	16.9706	1.32924	0.664619	+	0.747183i	$0.268594\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−15.3137	−1.18501	−0.592505	−	0.805567i	$-0.701861\pi$
$167$	−15.3137	−1.18501	−0.592505	+	0.805567i	$0.701861\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−8.48528	−0.634220	−0.317110	−	0.948389i	$-0.602712\pi$
$179$	−8.48528	−0.634220	−0.317110	+	0.948389i	$0.602712\pi$
$180$	0	0
$181$	19.6569	1.46108	0.730541	−	0.682869i	$-0.239268\pi$
$181$	19.6569	1.46108	0.730541	+	0.682869i	$0.239268\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−3.65685	−0.268857
$186$	0	0
$187$	3.31371	0.242322
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	0	0
$193$	−21.7990	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$193$	−21.7990	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.34315	0.164457
$204$	0	0
$205$	11.6569	0.814150
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	1.65685	0.112997
$216$	0	0
$217$	−4.68629	−0.318126
$218$	0	0
$219$	−3.17157	−0.214315
$220$	0	0
$221$	0.828427	0.0557260
$222$	0	0
$223$	−2.82843	−0.189405	−0.0947027	−	0.995506i	$-0.530190\pi$
$223$	−2.82843	−0.189405	−0.0947027	+	0.995506i	$0.530190\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	0	0
$229$	−8.82843	−0.583399	−0.291699	−	0.956510i	$-0.594221\pi$
$229$	−8.82843	−0.583399	−0.291699	+	0.956510i	$0.594221\pi$
$230$	0	0
$231$	11.3137	0.744387
$232$	0	0
$233$	26.4853	1.73511	0.867554	−	0.497343i	$-0.165691\pi$
$233$	26.4853	1.73511	0.867554	+	0.497343i	$0.165691\pi$
$234$	0	0
$235$	−9.65685	−0.629944
$236$	0	0
$237$	5.65685	0.367452
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	21.3137	1.37294	0.686468	−	0.727160i	$-0.259160\pi$
$241$	21.3137	1.37294	0.686468	+	0.727160i	$0.259160\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	15.3137	0.970467
$250$	0	0
$251$	−5.17157	−0.326427	−0.163213	−	0.986591i	$-0.552186\pi$
$251$	−5.17157	−0.326427	−0.163213	+	0.986591i	$0.552186\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	0.828427	0.0518781
$256$	0	0
$257$	−3.17157	−0.197837	−0.0989186	−	0.995096i	$-0.531538\pi$
$257$	−3.17157	−0.197837	−0.0989186	+	0.995096i	$0.531538\pi$
$258$	0	0
$259$	−10.3431	−0.642692
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	−14.1421	−0.872041	−0.436021	−	0.899937i	$-0.643613\pi$
$263$	−14.1421	−0.872041	−0.436021	+	0.899937i	$0.643613\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	0	0
$267$	−3.65685	−0.223796
$268$	0	0
$269$	−1.51472	−0.0923540	−0.0461770	−	0.998933i	$-0.514704\pi$
$269$	−1.51472	−0.0923540	−0.0461770	+	0.998933i	$0.514704\pi$
$270$	0	0
$271$	−9.65685	−0.586612	−0.293306	−	0.956019i	$-0.594755\pi$
$271$	−9.65685	−0.586612	−0.293306	+	0.956019i	$0.594755\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−12.6274	−0.758708	−0.379354	−	0.925252i	$-0.623854\pi$
$277$	−12.6274	−0.758708	−0.379354	+	0.925252i	$0.623854\pi$
$278$	0	0
$279$	−1.65685	−0.0991933
$280$	0	0
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	0	0
$283$	−25.6569	−1.52514	−0.762571	−	0.646905i	$-0.776063\pi$
$283$	−25.6569	−1.52514	−0.762571	+	0.646905i	$0.776063\pi$
$284$	0	0
$285$	2.82843	0.167542
$286$	0	0
$287$	32.9706	1.94619
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	1.51472	0.0887944
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	4.68629	0.270113
$302$	0	0
$303$	−6.48528	−0.372570
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	2.34315	0.133297
$310$	0	0
$311$	−30.6274	−1.73672	−0.868361	−	0.495933i	$-0.834826\pi$
$311$	−30.6274	−1.73672	−0.868361	+	0.495933i	$0.834826\pi$
$312$	0	0
$313$	10.9706	0.620093	0.310046	−	0.950721i	$-0.399655\pi$
$313$	10.9706	0.620093	0.310046	+	0.950721i	$0.399655\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	9.31371	0.523110	0.261555	−	0.965189i	$-0.415765\pi$
$317$	9.31371	0.523110	0.261555	+	0.965189i	$0.415765\pi$
$318$	0	0
$319$	−3.31371	−0.185532
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	2.34315	0.130376
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−2.48528	−0.137436
$328$	0	0
$329$	−27.3137	−1.50585
$330$	0	0
$331$	−7.51472	−0.413046	−0.206523	−	0.978442i	$-0.566215\pi$
$331$	−7.51472	−0.413046	−0.206523	+	0.978442i	$0.566215\pi$
$332$	0	0
$333$	−3.65685	−0.200394
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.6569	1.28867	0.644335	−	0.764743i	$-0.277134\pi$
$337$	23.6569	1.28867	0.644335	+	0.764743i	$0.277134\pi$
$338$	0	0
$339$	14.4853	0.786732
$340$	0	0
$341$	6.62742	0.358895
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−2.82843	−0.152277
$346$	0	0
$347$	1.65685	0.0889446	0.0444723	−	0.999011i	$-0.485839\pi$
$347$	1.65685	0.0889446	0.0444723	+	0.999011i	$0.485839\pi$
$348$	0	0
$349$	15.1716	0.812116	0.406058	−	0.913847i	$-0.366903\pi$
$349$	15.1716	0.812116	0.406058	+	0.913847i	$0.366903\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−2.68629	−0.142977	−0.0714884	−	0.997441i	$-0.522775\pi$
$353$	−2.68629	−0.142977	−0.0714884	+	0.997441i	$0.522775\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	2.34315	0.124012
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	3.17157	0.166008
$366$	0	0
$367$	−2.34315	−0.122311	−0.0611556	−	0.998128i	$-0.519479\pi$
$367$	−2.34315	−0.122311	−0.0611556	+	0.998128i	$0.519479\pi$
$368$	0	0
$369$	11.6569	0.606832
$370$	0	0
$371$	−26.3431	−1.36767
$372$	0	0
$373$	13.3137	0.689358	0.344679	−	0.938721i	$-0.387988\pi$
$373$	13.3137	0.689358	0.344679	+	0.938721i	$0.387988\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	−19.7990	−1.01701	−0.508503	−	0.861060i	$-0.669801\pi$
$379$	−19.7990	−1.01701	−0.508503	+	0.861060i	$0.669801\pi$
$380$	0	0
$381$	−2.34315	−0.120043
$382$	0	0
$383$	34.6274	1.76938	0.884689	−	0.466181i	$-0.154371\pi$
$383$	34.6274	1.76938	0.884689	+	0.466181i	$0.154371\pi$
$384$	0	0
$385$	−11.3137	−0.576600
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−32.1421	−1.62967	−0.814835	−	0.579692i	$-0.803173\pi$
$389$	−32.1421	−1.62967	−0.814835	+	0.579692i	$0.803173\pi$
$390$	0	0
$391$	−2.34315	−0.118498
$392$	0	0
$393$	10.8284	0.546222
$394$	0	0
$395$	−5.65685	−0.284627
$396$	0	0
$397$	4.34315	0.217976	0.108988	−	0.994043i	$-0.465239\pi$
$397$	4.34315	0.217976	0.108988	+	0.994043i	$0.465239\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−30.2843	−1.51232	−0.756162	−	0.654384i	$-0.772928\pi$
$401$	−30.2843	−1.51232	−0.756162	+	0.654384i	$0.772928\pi$
$402$	0	0
$403$	1.65685	0.0825338
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	14.6274	0.725054
$408$	0	0
$409$	−11.6569	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$409$	−11.6569	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$410$	0	0
$411$	−13.3137	−0.656717
$412$	0	0
$413$	−11.3137	−0.556711
$414$	0	0
$415$	−15.3137	−0.751720
$416$	0	0
$417$	2.34315	0.114744
$418$	0	0
$419$	−7.51472	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$419$	−7.51472	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$420$	0	0
$421$	4.82843	0.235323	0.117662	−	0.993054i	$-0.462460\pi$
$421$	4.82843	0.235323	0.117662	+	0.993054i	$0.462460\pi$
$422$	0	0
$423$	−9.65685	−0.469532
$424$	0	0
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	−5.65685	−0.273754
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	8.97056	0.432097	0.216048	−	0.976383i	$-0.430683\pi$
$431$	8.97056	0.432097	0.216048	+	0.976383i	$0.430683\pi$
$432$	0	0
$433$	−19.6569	−0.944648	−0.472324	−	0.881425i	$-0.656585\pi$
$433$	−19.6569	−0.944648	−0.472324	+	0.881425i	$0.656585\pi$
$434$	0	0
$435$	−0.828427	−0.0397200
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.9706	0.616250	0.308125	−	0.951346i	$-0.400298\pi$
$443$	12.9706	0.616250	0.308125	+	0.951346i	$0.400298\pi$
$444$	0	0
$445$	3.65685	0.173352
$446$	0	0
$447$	15.6569	0.740544
$448$	0	0
$449$	−15.6569	−0.738893	−0.369446	−	0.929252i	$-0.620453\pi$
$449$	−15.6569	−0.738893	−0.369446	+	0.929252i	$0.620453\pi$
$450$	0	0
$451$	−46.6274	−2.19560
$452$	0	0
$453$	−20.9706	−0.985283
$454$	0	0
$455$	−2.82843	−0.132599
$456$	0	0
$457$	−0.142136	−0.00664882	−0.00332441	−	0.999994i	$-0.501058\pi$
$457$	−0.142136	−0.00664882	−0.00332441	+	0.999994i	$0.501058\pi$
$458$	0	0
$459$	0.828427	0.0386677
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−6.14214	−0.285449	−0.142725	−	0.989762i	$-0.545586\pi$
$463$	−6.14214	−0.285449	−0.142725	+	0.989762i	$0.545586\pi$
$464$	0	0
$465$	1.65685	0.0768348
$466$	0	0
$467$	−15.3137	−0.708634	−0.354317	−	0.935125i	$-0.615287\pi$
$467$	−15.3137	−0.708634	−0.354317	+	0.935125i	$0.615287\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−21.3137	−0.982084
$472$	0	0
$473$	−6.62742	−0.304729
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−9.31371	−0.426445
$478$	0	0
$479$	10.3431	0.472590	0.236295	−	0.971681i	$-0.424067\pi$
$479$	10.3431	0.472590	0.236295	+	0.971681i	$0.424067\pi$
$480$	0	0
$481$	3.65685	0.166738
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	−1.51472	−0.0687798
$486$	0	0
$487$	−5.17157	−0.234346	−0.117173	−	0.993111i	$-0.537383\pi$
$487$	−5.17157	−0.234346	−0.117173	+	0.993111i	$0.537383\pi$
$488$	0	0
$489$	−16.9706	−0.767435
$490$	0	0
$491$	−7.51472	−0.339135	−0.169567	−	0.985519i	$-0.554237\pi$
$491$	−7.51472	−0.339135	−0.169567	+	0.985519i	$0.554237\pi$
$492$	0	0
$493$	−0.686292	−0.0309090
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−22.6274	−1.01498
$498$	0	0
$499$	−13.1716	−0.589641	−0.294820	−	0.955553i	$-0.595260\pi$
$499$	−13.1716	−0.589641	−0.294820	+	0.955553i	$0.595260\pi$
$500$	0	0
$501$	15.3137	0.684166
$502$	0	0
$503$	32.4853	1.44845	0.724224	−	0.689565i	$-0.242198\pi$
$503$	32.4853	1.44845	0.724224	+	0.689565i	$0.242198\pi$
$504$	0	0
$505$	6.48528	0.288591
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	1.31371	0.0582291	0.0291146	−	0.999576i	$-0.490731\pi$
$509$	1.31371	0.0582291	0.0291146	+	0.999576i	$0.490731\pi$
$510$	0	0
$511$	8.97056	0.396834
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	−2.34315	−0.103251
$516$	0	0
$517$	38.6274	1.69883
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−24.3431	−1.06649	−0.533246	−	0.845960i	$-0.679028\pi$
$521$	−24.3431	−1.06649	−0.533246	+	0.845960i	$0.679028\pi$
$522$	0	0
$523$	2.62742	0.114889	0.0574445	−	0.998349i	$-0.481705\pi$
$523$	2.62742	0.114889	0.0574445	+	0.998349i	$0.481705\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	1.37258	0.0597907
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−11.6569	−0.504914
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	8.48528	0.366167
$538$	0	0
$539$	−4.00000	−0.172292
$540$	0	0
$541$	11.4558	0.492525	0.246263	−	0.969203i	$-0.420797\pi$
$541$	11.4558	0.492525	0.246263	+	0.969203i	$0.420797\pi$
$542$	0	0
$543$	−19.6569	−0.843556
$544$	0	0
$545$	2.48528	0.106458
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−2.34315	−0.0998214
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	3.65685	0.155225
$556$	0	0
$557$	−27.9411	−1.18390	−0.591952	−	0.805973i	$-0.701642\pi$
$557$	−27.9411	−1.18390	−0.591952	+	0.805973i	$0.701642\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	−3.31371	−0.139905
$562$	0	0
$563$	−14.3431	−0.604492	−0.302246	−	0.953230i	$-0.597736\pi$
$563$	−14.3431	−0.604492	−0.302246	+	0.953230i	$0.597736\pi$
$564$	0	0
$565$	−14.4853	−0.609400
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	14.2843	0.598828	0.299414	−	0.954123i	$-0.403209\pi$
$569$	14.2843	0.598828	0.299414	+	0.954123i	$0.403209\pi$
$570$	0	0
$571$	−7.02944	−0.294173	−0.147086	−	0.989124i	$-0.546990\pi$
$571$	−7.02944	−0.294173	−0.147086	+	0.989124i	$0.546990\pi$
$572$	0	0
$573$	19.3137	0.806842
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	13.5147	0.562625	0.281313	−	0.959616i	$-0.409230\pi$
$577$	13.5147	0.562625	0.281313	+	0.959616i	$0.409230\pi$
$578$	0	0
$579$	21.7990	0.905935
$580$	0	0
$581$	−43.3137	−1.79696
$582$	0	0
$583$	37.2548	1.54294
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−6.34315	−0.261810	−0.130905	−	0.991395i	$-0.541788\pi$
$587$	−6.34315	−0.261810	−0.130905	+	0.991395i	$0.541788\pi$
$588$	0	0
$589$	4.68629	0.193095
$590$	0	0
$591$	−9.31371	−0.383115
$592$	0	0
$593$	4.34315	0.178352	0.0891758	−	0.996016i	$-0.471577\pi$
$593$	4.34315	0.178352	0.0891758	+	0.996016i	$0.471577\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	0	0
$597$	16.9706	0.694559
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−2.68629	−0.109576	−0.0547881	−	0.998498i	$-0.517448\pi$
$601$	−2.68629	−0.109576	−0.0547881	+	0.998498i	$0.517448\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−5.65685	−0.229605	−0.114802	−	0.993388i	$-0.536623\pi$
$607$	−5.65685	−0.229605	−0.114802	+	0.993388i	$0.536623\pi$
$608$	0	0
$609$	−2.34315	−0.0949491
$610$	0	0
$611$	9.65685	0.390675
$612$	0	0
$613$	−13.0294	−0.526254	−0.263127	−	0.964761i	$-0.584754\pi$
$613$	−13.0294	−0.526254	−0.263127	+	0.964761i	$0.584754\pi$
$614$	0	0
$615$	−11.6569	−0.470050
$616$	0	0
$617$	16.6274	0.669395	0.334697	−	0.942326i	$-0.391366\pi$
$617$	16.6274	0.669395	0.334697	+	0.942326i	$0.391366\pi$
$618$	0	0
$619$	34.8284	1.39987	0.699936	−	0.714205i	$-0.253212\pi$
$619$	34.8284	1.39987	0.699936	+	0.714205i	$0.253212\pi$
$620$	0	0
$621$	−2.82843	−0.113501
$622$	0	0
$623$	10.3431	0.414389
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−11.3137	−0.451826
$628$	0	0
$629$	3.02944	0.120792
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	2.34315	0.0929849
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−4.62742	−0.182772	−0.0913860	−	0.995816i	$-0.529130\pi$
$641$	−4.62742	−0.182772	−0.0913860	+	0.995816i	$0.529130\pi$
$642$	0	0
$643$	−14.6274	−0.576849	−0.288425	−	0.957503i	$-0.593131\pi$
$643$	−14.6274	−0.576849	−0.288425	+	0.957503i	$0.593131\pi$
$644$	0	0
$645$	−1.65685	−0.0652386
$646$	0	0
$647$	−41.4558	−1.62980	−0.814899	−	0.579603i	$-0.803207\pi$
$647$	−41.4558	−1.62980	−0.814899	+	0.579603i	$0.803207\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	4.68629	0.183670
$652$	0	0
$653$	38.2843	1.49818	0.749090	−	0.662469i	$-0.230491\pi$
$653$	38.2843	1.49818	0.749090	+	0.662469i	$0.230491\pi$
$654$	0	0
$655$	−10.8284	−0.423102
$656$	0	0
$657$	3.17157	0.123735
$658$	0	0
$659$	−4.20101	−0.163648	−0.0818241	−	0.996647i	$-0.526075\pi$
$659$	−4.20101	−0.163648	−0.0818241	+	0.996647i	$0.526075\pi$
$660$	0	0
$661$	4.82843	0.187804	0.0939020	−	0.995581i	$-0.470066\pi$
$661$	4.82843	0.187804	0.0939020	+	0.995581i	$0.470066\pi$
$662$	0	0
$663$	−0.828427	−0.0321734
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	2.34315	0.0907270
$668$	0	0
$669$	2.82843	0.109353
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	6.28427	0.241524	0.120762	−	0.992681i	$-0.461466\pi$
$677$	6.28427	0.241524	0.120762	+	0.992681i	$0.461466\pi$
$678$	0	0
$679$	−4.28427	−0.164415
$680$	0	0
$681$	6.34315	0.243070
$682$	0	0
$683$	−29.9411	−1.14567	−0.572833	−	0.819672i	$-0.694156\pi$
$683$	−29.9411	−1.14567	−0.572833	+	0.819672i	$0.694156\pi$
$684$	0	0
$685$	13.3137	0.508691
$686$	0	0
$687$	8.82843	0.336826
$688$	0	0
$689$	9.31371	0.354824
$690$	0	0
$691$	−34.8284	−1.32494	−0.662468	−	0.749090i	$-0.730491\pi$
$691$	−34.8284	−1.32494	−0.662468	+	0.749090i	$0.730491\pi$
$692$	0	0
$693$	−11.3137	−0.429772
$694$	0	0
$695$	−2.34315	−0.0888806
$696$	0	0
$697$	−9.65685	−0.365779
$698$	0	0
$699$	−26.4853	−1.00177
$700$	0	0
$701$	−9.51472	−0.359366	−0.179683	−	0.983725i	$-0.557507\pi$
$701$	−9.51472	−0.359366	−0.179683	+	0.983725i	$0.557507\pi$
$702$	0	0
$703$	10.3431	0.390099
$704$	0	0
$705$	9.65685	0.363698
$706$	0	0
$707$	18.3431	0.689865
$708$	0	0
$709$	−26.7696	−1.00535	−0.502676	−	0.864475i	$-0.667651\pi$
$709$	−26.7696	−1.00535	−0.502676	+	0.864475i	$0.667651\pi$
$710$	0	0
$711$	−5.65685	−0.212149
$712$	0	0
$713$	−4.68629	−0.175503
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	−16.9706	−0.633777
$718$	0	0
$719$	37.6569	1.40436	0.702182	−	0.711998i	$-0.252209\pi$
$719$	37.6569	1.40436	0.702182	+	0.711998i	$0.252209\pi$
$720$	0	0
$721$	−6.62742	−0.246818
$722$	0	0
$723$	−21.3137	−0.792665
$724$	0	0
$725$	0.828427	0.0307670
$726$	0	0
$727$	14.6274	0.542501	0.271250	−	0.962509i	$-0.412563\pi$
$727$	14.6274	0.542501	0.271250	+	0.962509i	$0.412563\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.37258	−0.0507668
$732$	0	0
$733$	1.02944	0.0380231	0.0190116	−	0.999819i	$-0.493948\pi$
$733$	1.02944	0.0380231	0.0190116	+	0.999819i	$0.493948\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−23.1127	−0.850214	−0.425107	−	0.905143i	$-0.639764\pi$
$739$	−23.1127	−0.850214	−0.425107	+	0.905143i	$0.639764\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−15.6569	−0.573623
$746$	0	0
$747$	−15.3137	−0.560299
$748$	0	0
$749$	11.3137	0.413394
$750$	0	0
$751$	51.3137	1.87246	0.936232	−	0.351383i	$-0.114288\pi$
$751$	51.3137	1.87246	0.936232	+	0.351383i	$0.114288\pi$
$752$	0	0
$753$	5.17157	0.188463
$754$	0	0
$755$	20.9706	0.763197
$756$	0	0
$757$	22.6863	0.824547	0.412274	−	0.911060i	$-0.364735\pi$
$757$	22.6863	0.824547	0.412274	+	0.911060i	$0.364735\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	14.9706	0.542682	0.271341	−	0.962483i	$-0.412533\pi$
$761$	14.9706	0.542682	0.271341	+	0.962483i	$0.412533\pi$
$762$	0	0
$763$	7.02944	0.254483
$764$	0	0
$765$	−0.828427	−0.0299518
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	35.9411	1.29607	0.648035	−	0.761610i	$-0.275591\pi$
$769$	35.9411	1.29607	0.648035	+	0.761610i	$0.275591\pi$
$770$	0	0
$771$	3.17157	0.114221
$772$	0	0
$773$	−53.3137	−1.91756	−0.958780	−	0.284148i	$-0.908289\pi$
$773$	−53.3137	−1.91756	−0.958780	+	0.284148i	$0.908289\pi$
$774$	0	0
$775$	−1.65685	−0.0595160
$776$	0	0
$777$	10.3431	0.371058
$778$	0	0
$779$	−32.9706	−1.18129
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	−0.828427	−0.0296056
$784$	0	0
$785$	21.3137	0.760719
$786$	0	0
$787$	42.3431	1.50937	0.754685	−	0.656087i	$-0.227790\pi$
$787$	42.3431	1.50937	0.754685	+	0.656087i	$0.227790\pi$
$788$	0	0
$789$	14.1421	0.503473
$790$	0	0
$791$	−40.9706	−1.45675
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	9.31371	0.330323
$796$	0	0
$797$	−12.6274	−0.447286	−0.223643	−	0.974671i	$-0.571795\pi$
$797$	−12.6274	−0.447286	−0.223643	+	0.974671i	$0.571795\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	3.65685	0.129209
$802$	0	0
$803$	−12.6863	−0.447690
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	1.51472	0.0533206
$808$	0	0
$809$	−25.3137	−0.889983	−0.444991	−	0.895535i	$-0.646793\pi$
$809$	−25.3137	−0.889983	−0.444991	+	0.895535i	$0.646793\pi$
$810$	0	0
$811$	15.1127	0.530679	0.265339	−	0.964155i	$-0.414516\pi$
$811$	15.1127	0.530679	0.265339	+	0.964155i	$0.414516\pi$
$812$	0	0
$813$	9.65685	0.338681
$814$	0	0
$815$	16.9706	0.594453
$816$	0	0
$817$	−4.68629	−0.163953
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	−14.2843	−0.498525	−0.249262	−	0.968436i	$-0.580188\pi$
$821$	−14.2843	−0.498525	−0.249262	+	0.968436i	$0.580188\pi$
$822$	0	0
$823$	−5.65685	−0.197186	−0.0985928	−	0.995128i	$-0.531434\pi$
$823$	−5.65685	−0.197186	−0.0985928	+	0.995128i	$0.531434\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	31.3137	1.08888	0.544442	−	0.838798i	$-0.316741\pi$
$827$	31.3137	1.08888	0.544442	+	0.838798i	$0.316741\pi$
$828$	0	0
$829$	41.3137	1.43488	0.717442	−	0.696618i	$-0.245313\pi$
$829$	41.3137	1.43488	0.717442	+	0.696618i	$0.245313\pi$
$830$	0	0
$831$	12.6274	0.438040
$832$	0	0
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	−15.3137	−0.529953
$836$	0	0
$837$	1.65685	0.0572693
$838$	0	0
$839$	−8.97056	−0.309698	−0.154849	−	0.987938i	$-0.549489\pi$
$839$	−8.97056	−0.309698	−0.154849	+	0.987938i	$0.549489\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	26.9706	0.928916
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	14.1421	0.485930
$848$	0	0
$849$	25.6569	0.880541
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	34.9706	1.19737	0.598685	−	0.800985i	$-0.295690\pi$
$853$	34.9706	1.19737	0.598685	+	0.800985i	$0.295690\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−3.17157	−0.108339	−0.0541694	−	0.998532i	$-0.517251\pi$
$857$	−3.17157	−0.108339	−0.0541694	+	0.998532i	$0.517251\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	−32.9706	−1.12363
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	16.3137	0.554043
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.51472	−0.0512655
$874$	0	0
$875$	2.82843	0.0956183
$876$	0	0
$877$	30.6863	1.03620	0.518101	−	0.855319i	$-0.326639\pi$
$877$	30.6863	1.03620	0.518101	+	0.855319i	$0.326639\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	0	0
$881$	10.9706	0.369608	0.184804	−	0.982775i	$-0.440835\pi$
$881$	10.9706	0.369608	0.184804	+	0.982775i	$0.440835\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	44.7696	1.50321	0.751607	−	0.659611i	$-0.229279\pi$
$887$	44.7696	1.50321	0.751607	+	0.659611i	$0.229279\pi$
$888$	0	0
$889$	6.62742	0.222276
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	27.3137	0.914018
$894$	0	0
$895$	−8.48528	−0.283632
$896$	0	0
$897$	2.82843	0.0944384
$898$	0	0
$899$	−1.37258	−0.0457782
$900$	0	0
$901$	7.71573	0.257048
$902$	0	0
$903$	−4.68629	−0.155950
$904$	0	0
$905$	19.6569	0.653416
$906$	0	0
$907$	49.6569	1.64883	0.824414	−	0.565987i	$-0.191505\pi$
$907$	49.6569	1.64883	0.824414	+	0.565987i	$0.191505\pi$
$908$	0	0
$909$	6.48528	0.215103
$910$	0	0
$911$	−8.97056	−0.297208	−0.148604	−	0.988897i	$-0.547478\pi$
$911$	−8.97056	−0.297208	−0.148604	+	0.988897i	$0.547478\pi$
$912$	0	0
$913$	61.2548	2.02724
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−30.6274	−1.01141
$918$	0	0
$919$	22.6274	0.746410	0.373205	−	0.927749i	$-0.378259\pi$
$919$	22.6274	0.746410	0.373205	+	0.927749i	$0.378259\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−3.65685	−0.120237
$926$	0	0
$927$	−2.34315	−0.0769590
$928$	0	0
$929$	−30.2843	−0.993595	−0.496797	−	0.867867i	$-0.665491\pi$
$929$	−30.2843	−0.993595	−0.496797	+	0.867867i	$0.665491\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	0	0
$933$	30.6274	1.00270
$934$	0	0
$935$	3.31371	0.108370
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−10.9706	−0.358011
$940$	0	0
$941$	−28.3431	−0.923960	−0.461980	−	0.886890i	$-0.652861\pi$
$941$	−28.3431	−0.923960	−0.461980	+	0.886890i	$0.652861\pi$
$942$	0	0
$943$	32.9706	1.07367
$944$	0	0
$945$	−2.82843	−0.0920087
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−3.17157	−0.102954
$950$	0	0
$951$	−9.31371	−0.302018
$952$	0	0
$953$	47.7401	1.54645	0.773227	−	0.634129i	$-0.218641\pi$
$953$	47.7401	1.54645	0.773227	+	0.634129i	$0.218641\pi$
$954$	0	0
$955$	−19.3137	−0.624977
$956$	0	0
$957$	3.31371	0.107117
$958$	0	0
$959$	37.6569	1.21600
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−21.7990	−0.701734
$966$	0	0
$967$	−3.79899	−0.122167	−0.0610836	−	0.998133i	$-0.519456\pi$
$967$	−3.79899	−0.122167	−0.0610836	+	0.998133i	$0.519456\pi$
$968$	0	0
$969$	−2.34315	−0.0752727
$970$	0	0
$971$	10.8284	0.347501	0.173750	−	0.984790i	$-0.444411\pi$
$971$	10.8284	0.347501	0.173750	+	0.984790i	$0.444411\pi$
$972$	0	0
$973$	−6.62742	−0.212465
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	41.5980	1.33084	0.665419	−	0.746470i	$-0.268253\pi$
$977$	41.5980	1.33084	0.665419	+	0.746470i	$0.268253\pi$
$978$	0	0
$979$	−14.6274	−0.467494
$980$	0	0
$981$	2.48528	0.0793489
$982$	0	0
$983$	−18.6274	−0.594122	−0.297061	−	0.954858i	$-0.596007\pi$
$983$	−18.6274	−0.594122	−0.297061	+	0.954858i	$0.596007\pi$
$984$	0	0
$985$	9.31371	0.296759
$986$	0	0
$987$	27.3137	0.869405
$988$	0	0
$989$	4.68629	0.149015
$990$	0	0
$991$	−13.6569	−0.433824	−0.216912	−	0.976191i	$-0.569599\pi$
$991$	−13.6569	−0.433824	−0.216912	+	0.976191i	$0.569599\pi$
$992$	0	0
$993$	7.51472	0.238472
$994$	0	0
$995$	−16.9706	−0.538003
$996$	0	0
$997$	−2.68629	−0.0850757	−0.0425379	−	0.999095i	$-0.513544\pi$
$997$	−2.68629	−0.0850757	−0.0425379	+	0.999095i	$0.513544\pi$
$998$	0	0
$999$	3.65685	0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bn.1.2	✓	2
4.3	odd	2		6240.2.a.bt.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bn.1.2	✓	2	1.1	even	1	trivial
6240.2.a.bt.1.1	yes	2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -0.828427 q^{17} -2.82843 q^{19} -2.82843 q^{21} +2.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.828427 q^{29} -1.65685 q^{31} +4.00000 q^{33} +2.82843 q^{35} -3.65685 q^{37} +1.00000 q^{39} +11.6569 q^{41} +1.65685 q^{43} +1.00000 q^{45} -9.65685 q^{47} +1.00000 q^{49} +0.828427 q^{51} -9.31371 q^{53} -4.00000 q^{55} +2.82843 q^{57} -4.00000 q^{59} -2.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} -2.82843 q^{69} -8.00000 q^{71} +3.17157 q^{73} -1.00000 q^{75} -11.3137 q^{77} -5.65685 q^{79} +1.00000 q^{81} -15.3137 q^{83} -0.828427 q^{85} -0.828427 q^{87} +3.65685 q^{89} -2.82843 q^{91} +1.65685 q^{93} -2.82843 q^{95} -1.51472 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} - 8 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 8 q^{31} + 8 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} - 8 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} - 8 q^{55} - 8 q^{59} - 4 q^{61} - 2 q^{65} - 16 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 - 8 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 8 * q^31 + 8 * q^33 + 4 * q^37 + 2 * q^39 + 12 * q^41 - 8 * q^43 + 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 - 8 * q^55 - 8 * q^59 - 4 * q^61 - 2 * q^65 - 16 * q^71 + 12 * q^73 - 2 * q^75 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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